Such a process is known from F. Mares, J. E. Galle, S. E. Diamond and F. J. Regina, Journal of Catalysis 112, (1988), pp. 145-156, according to which publication an alkane dinitrile, such as .alpha.,.omega.-butane dinitrile (succinonitrile), is hydrogenated in the presence of a catalyst consisting of finely divided rhodium halogenide on a magnesium oxide carrier. The catalyst is pretreated with sodium hydroxide, yielding metallic rhodium on a magnesium oxide carrier. During the hydrogenation reaction NH.sub.3 is supplied in an excess relative to the dinitrile. In the hydrogenation of succinonitrile, at a reaction temperature of 100.degree. C., a pressure of 50 atm (5 MPa) and a reaction time of 5.5 hours, a conversion of 89.4% and a selectivity to aminobutyronitrile of 87.3% is achieved. This corresponds to an aminobutyronitrile yield of 78%. Although this yield is high, a further increase in the yield is desirable. In addition to this, a large amount of undesirable by-product is formed (11.9% of the succinonitrile being converted into dimers and oligomers). The article also reports an .epsilon.-amino capronitrile yield of 66.6% from adiponitrile at 90% conversion. Dimers are formed in an amount of 2.7%.